Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4563 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(9,10,14,13)(11,17,19,15)(12,20,21,16)(18,22,24,23)(25,26,30,31)(27,32,28,33)(29,34,37,38)(35,39,36,40);(1,2,5,6)(3,7,8,4)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(26,31)(32,33)(34,38)(39,40);(2,6)(4,7)(10,13)(15,17)(16,20)(22,23)(25,27,30,28)(26,32,31,33)(29,36,37,35)(34,40,38,39);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,18)(13,17)(14,19)(16,22)(20,23)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(9,12,14,21)(10,16,13,20)(11,18,19,24)(15,22,17,23)(25,28,30,27)(26,33,31,32)(29,36,37,35)(34,40,38,39);(1,4)(2,3)(5,7)(6,8)(9,13,14,10)(11,17,19,15)(12,20,21,16)(18,23,24,22)(25,29,30,37)(26,34,31,38)(27,35,28,36)(32,39,33,40)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8088350);